Volume 1 / Part 4 — *183. THE SUM OF THE RELATION-NUMBERS OF A FIELD.
Reading edition
*183. THE SUM OF THE RELATION-NUMBERS OF A FIELD.
Summary of *183.
In this number we have to define and consider the sum of the relation-numbers of the members of , where is a relation of relations. Since relational sums are not commutative, we cannot define the sum of the relation-numbers of members of a class of relations : it is necessary that should be given as the field of a relation , where determines the order in which the summation is to be effected.
In order to avoid repetition, we replace by , so that if is a member of , is replaced by , i.e. by . This relation is like , and its field has no members in common with the field of , unless . Hence we are led to the following definition:
*183·01.
This definition is analogous to *112·01, as appears from *182·52, and the propositions of the present number are analogous to some of the propositions of *112.
We have not merely
*183·11.
but also
*183·15.
which is a proposition with a weaker hypothesis than that of *183·11 (cf. note to *182·34).
Important propositions in this number are
*183·13.
*183·2.
I.e. a sum is only zero when there is no summand except (at most) zero. (Cf. *162·4·45.)
*183·25.
*183-26.
This proposition connects addition and multiplication.
*183·31.
This proposition connects the two kinds of addition. We have also
*183·33.
The associative law of addition in a very general form is
*183·43.
Finally the connection of ordinal and cardinal addition is given by
*183·5.
*183·01.
*183·1.
*183·11.
Dem.
*183·12.
*183·13.
*183·14.
Dem.
*183·15.
*183·2.
Dem.
*183·22.
Dem.
*183·23.
*183·231.
Dem.
*183·24.
Dem.
*183·25.
Dem.
*183·26.
Dem.
*183·3.
*183·301.
*183·302.
Dem.
*183·31.
Dem.
*183·32.
Dem.
*183·33.
Dem.
*183·331.
*183·42.
Dem.
*183·43.
This is a form of the associative law of addition.
Dem.
*183·5.
Dem.